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G = D4.3Q16order 128 = 27

The non-split extension by D4 of Q16 acting via Q16/Q8=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: D4.3Q16, Q8.4SD16, C42.198C23, Q8⋊C823C2, D4⋊C8.4C2, C4⋊C4.28D4, (D4×Q8).1C2, C4.Q164C2, C4⋊C8.9C22, C4.18(C2×Q16), (C2×D4).253D4, (C4×C8).20C22, (C2×Q8).198D4, D42Q8.5C2, C4.SD162C2, C4.30(C2×SD16), C4⋊Q8.19C22, C4.10D810C2, C4.36(C8⋊C22), (C4×D4).29C22, (C4×Q8).29C22, C4.35(C8.C22), C2.18(D4.8D4), C22.164C22≀C2, C2.13(C22⋊SD16), C2.13(C22⋊Q16), (C2×C4).955(C2×D4), SmallGroup(128,369)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — D4.3Q16
C1C2C22C2×C4C42C4×Q8D4×Q8 — D4.3Q16
C1C22C42 — D4.3Q16
C1C22C42 — D4.3Q16
C1C22C22C42 — D4.3Q16

Generators and relations for D4.3Q16
 G = < a,b,c,d | a4=b2=c8=1, d2=a2c4, bab=dad-1=a-1, ac=ca, cbc-1=a-1b, dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 256 in 114 conjugacy classes, 38 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×8], C22, C22 [×4], C8 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×2], D4, Q8 [×2], Q8 [×9], C23, C42, C42, C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×3], C22×C4 [×3], C2×D4, C2×Q8, C2×Q8 [×8], C4×C8, D4⋊C4, Q8⋊C4 [×3], C4⋊C8 [×2], C4.Q8, C2.D8, C4×D4, C4×D4, C4×Q8, C22⋊Q8 [×3], C4⋊Q8 [×2], C4⋊Q8, C22×Q8, D4⋊C8, Q8⋊C8, C4.10D8, D42Q8, C4.Q16, C4.SD16, D4×Q8, D4.3Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, SD16 [×2], Q16 [×2], C2×D4 [×3], C22≀C2, C2×SD16, C2×Q16, C8⋊C22, C8.C22, C22⋊SD16, C22⋊Q16, D4.8D4, D4.3Q16

Character table of D4.3Q16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 111144222244488881644448888
ρ111111111111111111111111111    trivial
ρ211111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-11111-1-11-1-111-1-1-1-1-11111    linear of order 2
ρ41111-1-11111-1-11-1-11111111-1-1-1-1    linear of order 2
ρ51111111111-1-111-1-1-11-1-1-1-111-1-1    linear of order 2
ρ61111111111-1-111-1-1-1-11111-1-111    linear of order 2
ρ71111-1-11111111-11-1-1-1111111-1-1    linear of order 2
ρ81111-1-11111111-11-1-11-1-1-1-1-1-111    linear of order 2
ρ9222200-2-222-2-2-20200000000000    orthogonal lifted from D4
ρ10222200-2-22222-20-200000000000    orthogonal lifted from D4
ρ11222200-2-2-2-2002002-2000000000    orthogonal lifted from D4
ρ122222-2-222-2-200-22000000000000    orthogonal lifted from D4
ρ13222200-2-2-2-200200-22000000000    orthogonal lifted from D4
ρ1422222222-2-200-2-2000000000000    orthogonal lifted from D4
ρ152-2-222-22-200000000002-22-200-22    symplectic lifted from Q16, Schur index 2
ρ162-2-222-22-20000000000-22-22002-2    symplectic lifted from Q16, Schur index 2
ρ172-2-22-222-20000000000-22-2200-22    symplectic lifted from Q16, Schur index 2
ρ182-2-22-222-200000000002-22-2002-2    symplectic lifted from Q16, Schur index 2
ρ192-22-20000-22-22000000-2--2--2-2--2-200    complex lifted from SD16
ρ202-22-20000-222-2000000-2--2--2-2-2--200    complex lifted from SD16
ρ212-22-20000-22-22000000--2-2-2--2-2--200    complex lifted from SD16
ρ222-22-20000-222-2000000--2-2-2--2--2-200    complex lifted from SD16
ρ234-44-400004-40000000000000000    orthogonal lifted from C8⋊C22
ρ244-4-4400-44000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2544-4-4000000000000002i2i-2i-2i0000    complex lifted from D4.8D4
ρ2644-4-400000000000000-2i-2i2i2i0000    complex lifted from D4.8D4

Smallest permutation representation of D4.3Q16
On 64 points
Generators in S64
(1 33 53 59)(2 34 54 60)(3 35 55 61)(4 36 56 62)(5 37 49 63)(6 38 50 64)(7 39 51 57)(8 40 52 58)(9 18 43 31)(10 19 44 32)(11 20 45 25)(12 21 46 26)(13 22 47 27)(14 23 48 28)(15 24 41 29)(16 17 42 30)
(1 37)(2 50)(3 57)(4 8)(5 33)(6 54)(7 61)(9 27)(10 14)(11 24)(12 42)(13 31)(15 20)(16 46)(17 21)(18 47)(19 28)(22 43)(23 32)(25 41)(26 30)(29 45)(34 38)(35 51)(36 58)(39 55)(40 62)(44 48)(49 59)(52 56)(53 63)(60 64)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 31 49 22)(2 17 50 26)(3 29 51 20)(4 23 52 32)(5 27 53 18)(6 21 54 30)(7 25 55 24)(8 19 56 28)(9 37 47 59)(10 62 48 40)(11 35 41 57)(12 60 42 38)(13 33 43 63)(14 58 44 36)(15 39 45 61)(16 64 46 34)

G:=sub<Sym(64)| (1,33,53,59)(2,34,54,60)(3,35,55,61)(4,36,56,62)(5,37,49,63)(6,38,50,64)(7,39,51,57)(8,40,52,58)(9,18,43,31)(10,19,44,32)(11,20,45,25)(12,21,46,26)(13,22,47,27)(14,23,48,28)(15,24,41,29)(16,17,42,30), (1,37)(2,50)(3,57)(4,8)(5,33)(6,54)(7,61)(9,27)(10,14)(11,24)(12,42)(13,31)(15,20)(16,46)(17,21)(18,47)(19,28)(22,43)(23,32)(25,41)(26,30)(29,45)(34,38)(35,51)(36,58)(39,55)(40,62)(44,48)(49,59)(52,56)(53,63)(60,64), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,31,49,22)(2,17,50,26)(3,29,51,20)(4,23,52,32)(5,27,53,18)(6,21,54,30)(7,25,55,24)(8,19,56,28)(9,37,47,59)(10,62,48,40)(11,35,41,57)(12,60,42,38)(13,33,43,63)(14,58,44,36)(15,39,45,61)(16,64,46,34)>;

G:=Group( (1,33,53,59)(2,34,54,60)(3,35,55,61)(4,36,56,62)(5,37,49,63)(6,38,50,64)(7,39,51,57)(8,40,52,58)(9,18,43,31)(10,19,44,32)(11,20,45,25)(12,21,46,26)(13,22,47,27)(14,23,48,28)(15,24,41,29)(16,17,42,30), (1,37)(2,50)(3,57)(4,8)(5,33)(6,54)(7,61)(9,27)(10,14)(11,24)(12,42)(13,31)(15,20)(16,46)(17,21)(18,47)(19,28)(22,43)(23,32)(25,41)(26,30)(29,45)(34,38)(35,51)(36,58)(39,55)(40,62)(44,48)(49,59)(52,56)(53,63)(60,64), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,31,49,22)(2,17,50,26)(3,29,51,20)(4,23,52,32)(5,27,53,18)(6,21,54,30)(7,25,55,24)(8,19,56,28)(9,37,47,59)(10,62,48,40)(11,35,41,57)(12,60,42,38)(13,33,43,63)(14,58,44,36)(15,39,45,61)(16,64,46,34) );

G=PermutationGroup([(1,33,53,59),(2,34,54,60),(3,35,55,61),(4,36,56,62),(5,37,49,63),(6,38,50,64),(7,39,51,57),(8,40,52,58),(9,18,43,31),(10,19,44,32),(11,20,45,25),(12,21,46,26),(13,22,47,27),(14,23,48,28),(15,24,41,29),(16,17,42,30)], [(1,37),(2,50),(3,57),(4,8),(5,33),(6,54),(7,61),(9,27),(10,14),(11,24),(12,42),(13,31),(15,20),(16,46),(17,21),(18,47),(19,28),(22,43),(23,32),(25,41),(26,30),(29,45),(34,38),(35,51),(36,58),(39,55),(40,62),(44,48),(49,59),(52,56),(53,63),(60,64)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,31,49,22),(2,17,50,26),(3,29,51,20),(4,23,52,32),(5,27,53,18),(6,21,54,30),(7,25,55,24),(8,19,56,28),(9,37,47,59),(10,62,48,40),(11,35,41,57),(12,60,42,38),(13,33,43,63),(14,58,44,36),(15,39,45,61),(16,64,46,34)])

Matrix representation of D4.3Q16 in GL4(𝔽17) generated by

0100
16000
0010
0001
,
0100
1000
00160
00016
,
51200
5500
0020
0059
,
1000
01600
00162
00161
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[5,5,0,0,12,5,0,0,0,0,2,5,0,0,0,9],[1,0,0,0,0,16,0,0,0,0,16,16,0,0,2,1] >;

D4.3Q16 in GAP, Magma, Sage, TeX

D_4._3Q_{16}
% in TeX

G:=Group("D4.3Q16");
// GroupNames label

G:=SmallGroup(128,369);
// by ID

G=gap.SmallGroup(128,369);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,456,422,352,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=a^2*c^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;
// generators/relations

Export

Character table of D4.3Q16 in TeX

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